已知x2+3x+b≥13-1x2-3x(x∈R且x≠0)恒成立,则b的最小值为()A.-233B.5512C.133D.73-数学

题目简介

已知x2+3x+b≥13-1x2-3x(x∈R且x≠0)恒成立,则b的最小值为()A.-233B.5512C.133D.73-数学

题目详情

已知x2+3x+b≥
1
3
-
1
x2
-
3
x
(x∈R且x≠0)恒成立,则b的最小值为(  )
A.-
23
3
B.
55
12
C.
13
3
D.
7
3
题型:单选题难度:中档来源:不详

答案

x2+3x+b≥class="stub"1
3
-class="stub"1
x2
-class="stub"3
x
(x∈R且x≠0)恒成立,
可得b≥-x2-3x-class="stub"1
x2
-class="stub"3
x
+class="stub"1
3
,(x∈R且x≠0)恒成立,
求出-x2-3x-class="stub"1
x2
-class="stub"3
x
+class="stub"1
3
的最大值,
∵-x2-class="stub"1
x2
=-(x2+class="stub"1
x2
)≤-2,(x=1时等号成立);
-3x-class="stub"3
x
=-3(x+class="stub"1
x
)≤-6(x=1时等号成立);
∴-x2-3x-class="stub"1
x2
-class="stub"3
x
+class="stub"1
3
≤-2-6+class="stub"1
3
=-class="stub"23
3

∴b≥-class="stub"23
3

故选A;

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